This report contains three simulation studies for the Clusterwise Independent Vector Analysis (C-IVA) procedure
The first simulation is contains randomly generated data that is based on a CIVA model. This is a basic ‘concept of proof’ simulation. No special correlation structure between S across scanning sessions is imposed.
\[ X_i^{(D)} = P_{ir} S^{(D)}A_i^T\] Note that D represents a scanning session, other terms are similar to CICA. Note that S values are generated from a multivariate Laplace distribution.
The design is as follow:
fixed:
Size of the data are therefore: 2000 (V) X 50 (Ti) X 3 (D) X 40 subjects (R=2) or 80 subjects (R=4)
20 repetitions were used.
Note that the Ai matrices are non-square, which is not something that you encounter in the IVA literature. This was solved by computing:
\[ A_i^{(D)} = X_i^{(D)}S^{(D)T}(S^{(D)}S^{(D)T})^{\dagger} \]
Where \(\dagger\) is the Moore-Penrose pseudo inverse, this is a similar step that is done in (C-)ICA.
## Summary() of Adjusted Rand Index of clustering:## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 1 1 1 1 1 1## Summary() of percentage of random starts that yielded the lowest loss function value## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 0.7333 0.9000 0.9667 0.9488 1.0000 1.0000## Summary() of S Tucker congruence values:## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 0.6954 0.9903 0.9975 0.9706 0.9984 0.9997## Summary() of A Tucker congruence values:## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 0.6694 0.9971 0.9984 0.9728 0.9992 0.9999Congruence of S
## Err Q R Nr Stuck.mean
## 1 0.1 2 2 20 0.9978669
## 2 0.1 2 4 20 0.9991927
## 3 0.1 4 2 20 0.9697199
## 4 0.1 4 4 20 0.9833681
## 5 0.1 6 2 20 0.9105273
## 6 0.1 6 4 20 0.9484611
## 7 0.3 2 2 20 0.9990016
## 8 0.3 2 4 20 0.9976728
## 9 0.3 4 2 20 0.9890821
## 10 0.3 4 4 20 0.9886818
## 11 0.3 6 2 20 0.9257090
## 12 0.3 6 4 20 0.9412026
## 13 0.6 2 2 20 0.9962632
## 14 0.6 2 4 20 0.9978663
## 15 0.6 4 2 20 0.9909513
## 16 0.6 4 4 20 0.9912035
## 17 0.6 6 2 20 0.8998694
## 18 0.6 6 4 20 0.9438293Interactions S (first panel = 10% noise, middle = 30% noise, third panel 60% noise). X axis represents components, split by clusters (colours)
I guess CIVA robust against noise (as was the case for CICA), difficulties with underlying model complexity (Q) but unexpected results or R effect (more clusters seems to improve results).
Congruence of A
## R Q Nr Err Atuck.mean
## 1 2 2 20 0.1 0.9992391
## 2 2 2 20 0.3 0.9994163
## 3 2 2 20 0.6 0.9960557
## 4 2 4 20 0.1 0.9864100
## 5 2 4 20 0.3 0.9806526
## 6 2 4 20 0.6 0.9878275
## 7 2 6 20 0.1 0.8914611
## 8 2 6 20 0.3 0.9400483
## 9 2 6 20 0.6 0.8793592
## 10 4 2 20 0.1 0.9995025
## 11 4 2 20 0.3 0.9985120
## 12 4 2 20 0.6 0.9987863
## 13 4 4 20 0.1 0.9922047
## 14 4 4 20 0.3 0.9985119
## 15 4 4 20 0.6 0.9889148
## 16 4 6 20 0.1 0.9690951
## 17 4 6 20 0.3 0.9307943
## 18 4 6 20 0.6 0.9738912Interactions A (first panel = 10% noise, middle = 30% noise, third panel 60% noise). X axis represents components, split by clusters (colours)